Line Integrals

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as $W=\mathbf {F} \cdot \mathbf {s}$, have natural continuous analogues in terms of line integrals, in this case $\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s}$, which computes the work done on an object moving through an electric or gravitational field F along a path $L$.

Volume, path (line), and surface integrals

Volume Integrals

Given a scalar field $\rho :\mathbb {R} ^{3}\to \mathbb {R}$ that denotes a density at each specific point, and an arbitrary volume $\Omega \subseteq \mathbb {R} ^{3}$, the total "mass" $M$ inside of $\Omega$ can be determined by partitioning $\Omega$ into infinitesimal volumes. At each position $\mathbf {q} \in \Omega$, the volume of the infinitesimal volume is denoted by the infinitesimal $dV$. This gives rise to the following integral:

$M=\iiint _{\mathbf {q} \in \Omega }\rho (\mathbf {q} )dV$

Path Integrals

Given any oriented path $C$ (oriented means that there is a preferred direction), the differential $d\mathbf {q} =dx\mathbf {i} +dy\mathbf {j} +dz\mathbf {k}$ denotes an infinitesimal displacement along $C$ in the preferred direction. This differential can be used in various path integrals. Letting $f:\mathbb {R} ^{3}\to \mathbb {R}$ denote an arbitrary scalar field, and $\mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}$ denote an arbitrary vector field, various path integrals include:

$\int _{\mathbf {q} \in C}f(\mathbf {q} )d\mathbf {q}$, $\int _{\mathbf {q} \in C}f(\mathbf {q} )|d\mathbf {q} |$, $\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$, $\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )|d\mathbf {q} |$, and many more.

$\int _{\mathbf {q} \in C}d\mathbf {q}$ denotes the total displacement along $C$, and $\int _{\mathbf {q} \in C}|d\mathbf {q} |$ denotes the total length of $C$.

Calculating Path Integrals

To compute a path integral, the continuous oriented curve $C$ must be parameterized. $\mathbf {q} _{C}(t)$ will denote the point along $C$ indexed by $t$ from the range $[t_{0},t_{1}]$. $\mathbf {q} _{C}(t_{0})=\mathbf {q} _{0}$ must be the starting point of $C$ and $\mathbf {q} _{C}(t_{1})=\mathbf {q} _{1}$ must be the ending point of $C$. As $t$ increases, $\mathbf {q} _{C}(t)$ must proceed along $C$ in the preferred direction. An infinitesimal change in $t$, $dt$, results in the infinitesimal displacement $d\mathbf {q} ={\frac {d\mathbf {q} _{C}}{dt}}dt$ along $C$. In the path integral $\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$, the differential $d\mathbf {q}$ can be replaced with ${\frac {d\mathbf {q} _{C}}{dt}}dt$ to get $\int _{t=t_{0}}^{t_{1}}\mathbf {F} (\mathbf {q} _{C}(t))\cdot {\frac {d\mathbf {q} _{C}}{dt}}dt$

Example 1: As an example, consider the vector field $\mathbf {F} (x,y,z)=3\mathbf {i} -x\mathbf {j} +5y\mathbf {k}$ and the straight line curve $C$ that starts at $(1,1,1)$ and ends at $(7,-1,-2)$. $C$ can be parameterized by $\mathbf {q} _{C}(t)=(1+6t,1-2t,1-3t)$ where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in [0, 1]} . ${\frac {d\mathbf {q} _{C}}{dt}}=6\mathbf {i} -2\mathbf {j} -3\mathbf {k}$. We can then evaluate the path integral $\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$ as follows:

$\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\int _{\mathbf {q} \in C}(3\mathbf {i} -x\mathbf {j} +5y\mathbf {k} )\cdot d\mathbf {q} =\int _{t=0}^{1}(3\mathbf {i} -(1+6t)\mathbf {j} +5(1-2t)\mathbf {k} )\cdot (6\mathbf {i} -2\mathbf {j} -3\mathbf {k} )dt$

$=\int _{t=0}^{1}(18+(2+12t)+(-15+30t))dt=\int _{t=0}^{1}(5+42t)dt=(5t+21t^{2}){\bigg |}_{t=0}^{1}=26$

If a vector field $\mathbf {F}$ denotes a "force field", which returns the force on an object as a function of position, the work performed on a point mass that traverses the oriented curve $C$ is $W=\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$

Example 2: Consider the gravitational field that surrounds a point mass of $M$ located at the origin: $\mathbf {g} (\mathbf {q} )=-{\frac {GM}{|\mathbf {q} |^{2}}}{\frac {\mathbf {q} }{|\mathbf {q} |}}$ using Newton's inverse square law. The force acting on a point mass of $m$ at position $\mathbf {q}$ is $\mathbf {F} (\mathbf {q} )=m\mathbf {g} (\mathbf {q} )=-{\frac {GMm}{|\mathbf {q} |^{2}}}{\frac {\mathbf {q} }{|\mathbf {q} |}}$. In spherical coordinates the force is $\mathbf {F} (r,\theta ,\phi )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}$ (note that ${\hat {\mathbf {r} }},{\hat {\mathbf {\theta } }},{\hat {\mathbf {\phi } }}$ are the unit length mutually orthogonal basis vectors for spherical coordinates).

Consider an arbitrary path $C$ that $m$ traverses that starts at an altitude of $r=r_{1}$ and ends at an altitude of $r=r_{2}$. The work done by the gravitational field is:

$W=\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q}$

The infinitesimal displacement $d\mathbf {q}$ is equivalent to the displacement expressed in spherical coordinates: $dr\cdot {\hat {\mathbf {r} }}+r\cdot d\theta \cdot {\hat {\mathbf {\theta } }}+r\sin \theta \cdot d\phi \cdot {\hat {\mathbf {\phi } }}$.

$W=\int _{\mathbf {q} \in C}-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}\cdot (dr\cdot {\hat {\mathbf {r} }}+r\cdot d\theta \cdot {\hat {\mathbf {\theta } }}+r\sin \theta \cdot d\phi \cdot {\hat {\mathbf {\phi } }})=\int _{r=r_{1}}^{r_{2}}-{\frac {GMm}{r^{2}}}dr={\frac {GMm}{r}}{\bigg |}_{r=r_{1}}^{r_{2}}=GMm\left({\frac {1}{r_{2}}}-{\frac {1}{r_{1}}}\right)$

The work is equal to the amount of gravitational potential energy lost, so one possible function for the gravitational potential energy is $\phi (r,\theta ,\phi )=-{\frac {GMm}{r}}$ or equivalently, $\phi (\mathbf {q} )=-{\frac {GMm}{|\mathbf {q} |}}$.

Example 3: Consider the spiral $C$ parameterized with respect to $t$ in cylindrical coordinates by $\mathbf {q} _{C}(t)=(\rho =R,\phi =t,z=k{\frac {t}{2\pi }})$. Consider the problem of determining the spiral's length with $t$ restricted to the range $[0,2\pi ]$. An infinitesimal change of $dt$ in $t$ results in the infinitesimal displacement:

$d\mathbf {q} ={\frac {d\mathbf {q} _{C}}{dt}}dt=\left({\frac {d\rho }{dt}}{\hat {\mathbf {\rho } }}+\rho {\frac {d\phi }{dt}}{\hat {\mathbf {\phi } }}+{\frac {dz}{dt}}{\hat {\mathbf {z} }}\right)dt=\left(R{\hat {\mathbf {\phi } }}+{\frac {k}{2\pi }}{\hat {\mathbf {z} }}\right)dt$

The length of the infinitesimal displacement is $|d\mathbf {q} |={\sqrt {R^{2}+\left({\frac {k}{2\pi }}\right)^{2}}}\cdot dt$.

The length of the spiral is therefore: $\int _{\mathbf {q} \in C}|d\mathbf {q} |=\int _{t=0}^{2\pi }{\sqrt {R^{2}+\left({\frac {k}{2\pi }}\right)^{2}}}\cdot dt=2\pi {\sqrt {R^{2}+\left({\frac {k}{2\pi }}\right)^{2}}}={\sqrt {(2\pi R)^{2}+k^{2}}}$

Path Independence

If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is,

$\mathbf {F} =\nabla G,$

then by the multivariable chain rule, the derivative of the composition of G and r(t) is

${\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)$

which happens to be the integrand for the line integral of F on r(t). It follows, given a path C , that

$\int _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).$

In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.

Surface Integrals

Given any oriented surface $\sigma$ (oriented means that the there is a preferred direction to pass through the surface), an infinitesimal portion of the surface is defined by an infinitesimal area $|dS|$, and a unit length outwards oriented normal vector $\mathbf {n}$. $\mathbf {n}$ has a length of 1 and is perpendicular to the surface of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} , while penetrating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} in the preferred direction. The infinitesimal portion of the surface is denoted by the infinitesimal "surface vector": Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = |dS|\mathbf{n}} . If a vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}: \R^3 \to \R^3} denotes a flow density, then the flow through the infinitesimal surface portion in the preferred direction is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}} .

The infinitesimal "surface vector" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = \mathbf{n}|dS|} describes the infinitesimal surface element in a manner similar to how the infinitesimal displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathbf{q}} describes an infinitesimal portion of a path. More specifically, similar to how the interior points on a path do not affect the total displacement, the interior points on a surface to not affect the total surface vector.

The displacement between two points is independent of the path that connects them.

Consider for instance two paths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} and $C_{2}$ that both start at point $A$, and end at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . The total displacements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_1} d\mathbf{q}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_2} d\mathbf{q}} , are both equivalent and equal to the displacement between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . Note however that the total lengths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_1} |d\mathbf{q}|} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C_2} |d\mathbf{q}|} are not necessarily equivalent.

Similarly, given two surfaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} that both share the same counter-clockwise oriented boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , the total surface vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} \mathbf{dS}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_2} \mathbf{dS}} are both equivalent and are a function of the boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} . This implies that a surface can be freely deformed within its boundaries without changing the total surface vector. Note however that the surface areas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} |\mathbf{dS}|} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_2} |\mathbf{dS}|} are not necessarily equivalent.

The fact that the total surface vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} are equivalent is not immediately obvious. To prove this fact, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} be a constant vector field. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} share the same boundary, so the flux/flow of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} is equivalent. The flux through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_1 = \int_{\mathbf{q} \in S_1} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} } , and similarly for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_2 = \int_{\mathbf{q} \in S_2} \mathbf{F} \cdot \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} } . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} \cdot \int_{\mathbf{q} \in S_1} \mathbf{dS} = \mathbf{F} \cdot \int_{\mathbf{q} \in S_2} \mathbf{dS} } for every choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} , it follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in S_1} \mathbf{dS} = \int_{\mathbf{q} \in S_2} \mathbf{dS} } .

The geometric significance of the total surface vector is that each component measures the area of the projection of the surface onto the plane formed by the other two dimensions. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} be a surface with surface vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S} = S_x\mathbf{i} + S_y\mathbf{j} + S_z\mathbf{k}} . It is then the case that: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_x} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the yz-plane; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_y} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the xz-plane; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_z} is the area of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} onto the xy-plane.

The boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Sigma} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} is counter-clockwise oriented.

Given an oriented surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} , another important concept is the oriented boundary. The boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} is an oriented curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Sigma} but how is the orientation chosen? If the boundary is "counter-clockwise" oriented, then the boundary must follow a counter-clockwise direction when the oriented surface normal vectors point towards the viewer. The counter-clockwise boundary also obeys the "right-hand rule": If you hold your right hand with your thumb in the direction of the surface normals (penetrating the surface in the "preferred" direction), then your fingers will wrap around in the direction of the counter-clockwise oriented boundary.

Example 1: Consider the Cartesian points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0,0)} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0,0)} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1,0)} ; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0,1)} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1} be the surface formed by the triangular planes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(0,1,0), (0,0,0), (1,0,0)\}} ; $\{(0,0,1),(0,0,0),(0,1,0)\}$; and $\{(1,0,0),(0,0,0),(0,0,1)\}$ where the vertices are listed in a counterclockwise direction relative to the surface normal directions. The surface vectors of each plane are respectively ${\frac {1}{2}}\mathbf {k}$; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\mathbf{i}} ; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\mathbf{j}} respectively which add to a total surface vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S}_1 = \frac{1}{2}(\mathbf{i}+\mathbf{j}+\mathbf{k})} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_2} be the surface formed by the single triangular plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(1,0,0), (0,1,0), (0,0,1)\}} where the vertices are listed in a counterclockwise direction relative to the normal direction. It can be seen that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_2} share a the common counter clockwise boundary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0,0) \to (0,1,0) \to (0,0,1)} The surface vector is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S}_2 = \frac{1}{2}(\mathbf{j} - \mathbf{i}) \times (\mathbf{k} - \mathbf{i}) = \frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k})} which is equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S}_1} .

Example 2: This example will show how moving a point that is in the interior of a "triangular mesh" does not affect the total surface vector. Consider the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0, P_1, P_2, \dots, P_n} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq 3} . Let the closed path Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} be defined by the cycle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1 \to P_2 \to \dots \to P_n \to P_1} . For simplicity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{n+1} = P_1} . For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, 2, \dots, n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_i} will denote the displacement of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_i} relative to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0} . Like with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{n+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{n+1} = \mathbf{v}_1} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} denote a surface that is a "triangular mesh" comprised of the closed fan of triangles: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P_2,P_0,P_1\}} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P_3,P_0,P_2\}} ; ...; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P_n,P_0,P_{n-1}\}} ; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P_1,P_0,P_n\}} where the vertices of each triangle are listed in a counterclockwise direction. It can be seen that the counterclockwise boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} and does not depend on the location of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0} . The total surface vector for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S} = \frac{1}{2}(\mathbf{v}_1 \times \mathbf{v}_2) + \frac{1}{2}(\mathbf{v}_2 \times \mathbf{v}_3) + \dots + \frac{1}{2}(\mathbf{v}_{n-1} \times \mathbf{v}_n) + \frac{1}{2}(\mathbf{v}_n \times \mathbf{v}_{n+1}) = \frac{1}{2}\sum_{i=1}^n (\mathbf{v}_i \times \mathbf{v}_{i+1})}

Now displace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{w}} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'_0} . The displacement vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_i} relative to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0'} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}'_i = \mathbf{v}_i - \mathbf{w}} . The counterclockwise boundary is unaffected. The total surface vector is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S}' = \frac{1}{2}\sum_{i=1}^n (\mathbf{v}'_i \times \mathbf{v}'_{i+1}) = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i - \mathbf{w}) \times (\mathbf{v}_{i+1} - \mathbf{w})) = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i \times \mathbf{v}_{i+1}) - (\mathbf{v}_i \times \mathbf{w}) - (\mathbf{w} \times \mathbf{v}_{i+1}) + (\mathbf{w} \times \mathbf{w})) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{2}\sum_{i=1}^n ((\mathbf{v}_i \times \mathbf{v}_{i+1}) - \mathbf{w} \times (\mathbf{v}_{i+1} - \mathbf{v}_i)) = \mathbf{S} - \frac{1}{2}\mathbf{w} \times (\mathbf{v}_{n+1} - \mathbf{v}_1) = \mathbf{S} - \frac{1}{2}\mathbf{w} \times \mathbf{0} = \mathbf{S}}

Therefore moving the interior point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0} neither affects the boundary, nor the total surface vector.

Calculating Surface Integrals

To calculate a surface integral, the oriented surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} must be parameterized. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u, v)} be a continuous function that maps each point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v)} from a two-dimensional domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{u,v}} to a point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u, v)} must be continuous and onto. While $\mathbf {q} _{\sigma }(u,v)$ does not necessarily have to be one to one, the parameterization should never "fold back" on itself. The infinitesimal increases in $u$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} are respectively Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dv} . These respectively give rise to the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial u}du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv} . Assuming that the surface's orientation follows the right hand rule with respect to the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial u}du} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \mathbf{q}_{\sigma}}{\partial v}dv} , the surface vector that arises is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} .

In the surface integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_{\mathbf{q} \in \sigma} \mathbf{F}(\mathbf{q}) \cdot \mathbf{dS}} , the differential Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS}} can be replaced with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_{(u,v) \in D_{u,v}} \mathbf{F}(\mathbf{q}_{\sigma}(u,v)) \cdot (\frac{\partial \mathbf{q}_{\sigma}}{\partial u} \times \frac{\partial \mathbf{q}_{\sigma}}{\partial v})dudv} .

Example 3: Consider the problem of computing the surface area of a sphere of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

Center the sphere Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} on the origin, and using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} as the parameter variables, the sphere can be parameterized in spherical coordinates via Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{\sigma}(u,v) = (r = R, \theta = u, \phi = v)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \in [0, \pi]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in [-\pi, +\pi]} . The infinitesimal displacements from small changes in the parameters are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du} causes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial\mathbf{q}_{\sigma}}{\partial u}du = (\frac{\partial r}{\partial u}\hat{\mathbf{r}} + r\frac{\partial \theta}{\partial u}\hat{\mathbf{\theta}} + r\sin\theta\frac{\partial \phi}{\partial u}\hat{\mathbf{\phi}})du = (R\hat{\mathbf{\theta}})du}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dv} causes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial\mathbf{q}_{\sigma}}{\partial v}dv = (\frac{\partial r}{\partial v}\hat{\mathbf{r}} + r\frac{\partial \theta}{\partial v}\hat{\mathbf{\theta}} + r\sin\theta\frac{\partial \phi}{\partial v}\hat{\mathbf{\phi}})dv = (R\sin(u)\hat{\mathbf{\phi}})dv}

The infinitesimal surface vector is hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{dS} = (R\hat{\mathbf{\theta}})du \times (R\sin(u)\hat{\mathbf{\phi}})dv = (R^2\sin(u)\hat{\mathbf{r}})dudv} . While not important to this example, note how the parameterization was chosen so that the surface vector points outwards. The area is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{dS}| = R^2\sin(u)dudv} .

The total surface area is hence:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_{\mathbf{q} \in \sigma} |\mathbf{dS}| = \int_{u = 0}^{\pi}\int_{v = -\pi}^{+\pi} R^2\sin(u)dvdu = 2\pi R^2\int_{u = 0}^{\pi} \sin(u)du = 2\pi R^2 (-\cos(u)\bigg|_{u=0}^{\pi}) = 4\pi R^2}

More on Surface Integrals and Area

Resources

• Line Integrals, Mathematics LibreTexts
• Vector Calculus, Wikibooks: Calculus
• Line integral, Wikipedia
• Introduction to The Line Integral, Khan Academy

Licensing

Content obtained and/or adapted from:

• Vector calculus, Wikibooks: Calculus under a CC BY-SA license